The algorithms () and ()
use constant parameters \(\sigma \)  and \(\tau \) .
The stepsize condition (REF ) involves
the matrix norm \(\Vert A\Vert \)  and the Lipschitz constant \(L\) 
in (REF ).
Estimating or bounding \(\Vert A\Vert \)  for a large matrix can be difficult.
As an added complication, the norms \(\Vert \cdot \Vert _\mathrm {p}\)  and
\(\Vert \cdot \Vert _\mathrm {d}\)  in the definition of the matrix norm (REF )
are assumed to be scaled
so that the strong convexity parameters of the primal and dual kernels
are equal to one. Close bounds on the strong convexity parameters
may also be difficult to obtain.
Using conservative bounds for \(\Vert A\Vert \)  and \(L\) 
results in unnecessarily small values of \(\sigma \)  and \(\tau \) ,
and can dramatically slow down the convergence.
Even when the estimates of \(\Vert A\Vert \)  and \(L\)  are accurate,
the requirements for the stepsizes (REF ) are still
too strict in most iterations, as observed in [1]}.
In view of the above arguments,
line search techniques for primal–dual proximal methods have recently
become an active area of research.
Malitsky and Pock [2]} proposed a line search technique
for PDHG and the Condat–Vũ algorithm in the Euclidean case.
The algorithm with adaptive parameters in [3]}
focuses on a special case of (REF ) (i.e., \(f=0\) )
and extends the Loris–Verhoeven algorithm (REF ).
A Bregman proximal splitting method with line search
is discussed in [4]}
and considers the problem (REF ) with \(h=0\)  and \(g=\delta _{\lbrace b\rbrace }\) .
In this section, we extend the Bregman dual Condat–Vũ
algorithm () with a varying parameter option,
in which the stepsizes are chosen adaptively without requiring
any estimates or bounds for \(\Vert A\Vert \)  or the strong convexity parameter
of the kernels.
The algorithm is restricted to problems in the equality constrained
form
\( \begin{array}{ll}\mbox{minimize} & f(x) + h(x) \\\mbox{subject to} & Ax=b.\end{array}\) 
